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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 209814cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 209814.h4 | 209814cp1 | \([1, 1, 0, -204464471, 879932947029]\) | \(22106889268753393/4969545596928\) | \(212503613290108702741757952\) | \([2]\) | \(92897280\) | \(3.7647\) | \(\Gamma_0(N)\)-optimal |
| 209814.h2 | 209814cp2 | \([1, 1, 0, -3069124951, 65438494456405]\) | \(74768347616680342513/5615307472896\) | \(240117150441881250845835264\) | \([2, 2]\) | \(185794560\) | \(4.1113\) | |
| 209814.h1 | 209814cp3 | \([1, 1, 0, -49105114071, 4188283572076245]\) | \(306234591284035366263793/1727485056\) | \(73869292301411268376704\) | \([2]\) | \(371589120\) | \(4.4578\) | |
| 209814.h3 | 209814cp4 | \([1, 1, 0, -2867703511, 74398646646229]\) | \(-60992553706117024753/20624795251201152\) | \(-881940497126787847882109040768\) | \([2]\) | \(371589120\) | \(4.4578\) |
Rank
sage: E.rank()
The elliptic curves in class 209814cp have rank \(0\).
Complex multiplication
The elliptic curves in class 209814cp do not have complex multiplication.Modular form 209814.2.a.cp
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
